Mr. Hewitt, Mrs. Hoffman, and Mrs. Chisholm are engaged in a laser tag battle. Their lasers have a maximum range of 16 feet. This is the final showdown and whoever takes Mr. Hewitt out this round will forever be

I had a question about a problem that I was working on for my pre-calculus class.

Here's the problem statement:

The area of the parallelogram with vertices 0, $м$, $w$, and $v+w$ is 34. Find the area of the parallelogram with vertices 0, $Av$, $Aw$, and $Av+Aw$, where
$A=\left(\begin{array}{cc}3& -5\\ -1& -3\end{array}\right).$
I got the answer by doing something very tedious. I set $v=(\genfrac{}{}{0ex}{}{17}{0})$ and $w=(\genfrac{}{}{0ex}{}{0}{2})$, and did some really crazy matrix multiplication and a lot of plotting points of GeoGebra to get the answer of: $\boxed{{\displaystyle 476}}$.

Now, I'm 100% sure that was not the fastest way, can someone tell me the non-bash way to do the problem?

Finding explicit formula for intersection of perpendicular bisectors of a triangle

Using Bertrand's postulate which states:
For every integer $n\ge 1$ there is a prime number p such that $n<p\le 2n$
Prove that there exists infinitely many primes whose decimal expansion starts with 1.
I'm guessing I need to use something equal to 10 in this as it is a decimal expansion, but I'm not seeing where to start?
Any guidance would be great, thank you

If the non-base angle of angle of anisosceles triangle has a measure of 70o, what is themeasure of each base angle?
How many diagonals does a decagon have?

"The sum of the squares of the diagonals is equal to the sum of the squares of the four sides of a parallelogram."
I find this property very useful while solving different problems on Quadrilaterals & Polygon,so I am very inquisitive about a intuitive proof of this property.

Show that the orthocenter of ABC lies on $P^{\prime }{Q}^{\prime }$ , where $P^{\prime },Q^{\prime },R^{\prime }$ are the symmetric points of M to this sides of the triangle, M on circumscribed

If a line L separates a parallelogram into two regions of equal areas, then L contains the point of intersection of the diagonals of the parallelogram.
The figure shows a line L horizontally through the sides of the parallelogram.
This creates two trapezoids and I can intuitively show that if the bases of the trapezoids are not congruent then the areas can not be equal.
I just can not currently see how to relate the intersection with the diagonals.
Any help will be appreciated.